A266254 Binary representation of the n-th iteration of the "Rule 11" elementary cellular automaton starting with a single ON (black) cell.

1, 100, 11, 1111100, 11, 11111111100, 11, 111111111111100, 11, 1111111111111111100, 11, 11111111111111111111100, 11, 111111111111111111111111100, 11, 1111111111111111111111111111100, 11, 11111111111111111111111111111111100, 11

Offset: 0

Author

Robert Price, Dec 25 2015

Keywords

• nonn
• easy

Comments

Rule 43 also generates this sequence.

Links

• Robert Price, Table of n, a(n) for n = 0..999
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata
• Index entries for linear recurrences with constant coefficients, signature (0,10001,0,-10000).

Crossrefs

Cf. A266253, A266255.

Programs

Mathematica

rule=11; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]]],{k,1,rows}]   (* Binary Representation of Rows *)

Python

print([(10*100**n - 100)//9 if n%2 else 11 - 10*0**n for n in range(50)]) # Karl V. Keller, Jr., Aug 26 2021

Formula

From Colin Barker, Dec 26 2015 and Apr 14 2019: (Start)

a(n) = (199*(-1)^n+10^(2*n+1)-(-1)^n*10^(2*n+1)-1)/18 for n>0.

a(n) = 10001*a(n-2) - 10000*a(n-4) for n>4.

G.f.: (1+100*x-9990*x^2+111000*x^3-100000*x^4) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)).

(End)

a(n) = (10*100^n - 100)/9 for odd n; a(n) = 11 - 10*0^n for even n. - Karl V. Keller, Jr., Aug 26 2021